Cork Institute of Technology
Bachelor of Engineering in Civil Engineering - Award
(NFQ – Level 7)
Autumn 2009
Mathematics & Computing – old syllabus
(Time: 3 Hours)
Any FIVE questions are to be answered.
Examiners:
Mr J Lapthorne
Mr J Kindregan
Ms J English
Mr T O Leary
Section A
1 (a) List the order of precedence of the arithmetic operators.
(2 marks)
(b) What is the output of the following Fortran90 program?
NOTE: Trace your working so that partial credit may be given for incomplete or
incorrect work.
PROGRAM q1_b
implicit none
integer::limit = 6
integer::a, b
do a = limit,0, -1
do b = limit, 0, -1
if( (0 == a) .or. (limit == a) )then
write(*,'(i1)',advance='no')a
else if( (b - a == 0))then
write(*,'(i1)',advance='no')a
else if( (b + a == limit))then
write(*,'(i1)',advance='no')b
else
write(*,'(a)',advance='no')'#'
end if
end do
(8 marks)
(c) Write a program that will read in the coordinates of two points ( x1 , y1 ) and ( x 2 , y 2 ) and
calculate the distance between them. The program should allow more than one set of data to
be entered if the user wishes.
Note: The distance between the two points (x1,y1) and (x2,y2) is given by
(x1 -x 2 ) 2 +(y1 -y 2 ) 2
and the sqrt( ) function returns the square root of a real.
(10 marks)
2 (a) What is the output of the following Fortran90 program?
NOTE: Trace your working so that partial credit may be given for incomplete or
incorrect work.
PROGRAM q2_a
implicit none
integer::thisOne, thatOne
thisOne = 12
thatOne = 2
do
write(*, '(i5)', advance= 'no')thisOne
if(mod(thatOne, 5) == 0)then
print*
end if
if(thisOne == 1)exit
if(mod(thisOne, 2) == 0)then
thisOne = thisOne / 2
else
thisOne = thisOne * 3 + 1
end if
thatOne = thatOne + 1
dd
(8
marks)
2
(b)
Write a menu-driven program that allows the user to convert measurements either from
miles to kilometres (1 mile=1.60935 km), from feet to meters ( 1 foot=0.3048m), or
5
from degrees Fahrenheit to degrees Celsius {C= (F-32)}. A sample run of the program
9
should proceed as follows:
Available options are:
0.
Display this menu.
1.
Convert miles to kilometres.
2.
Convert feet to meters.
3.
Convert degrees Fahrenheit to degrees Celsius.
4.
Quit.
Enter an option (0 to see menu): 3
Enter degrees Fahrenheit: 212
This is equivalent to 100.00 degrees Celsius.
Enter an option (0 to see menu): 0
Available options are:
1.
Display this menu.
2.
Convert miles to kilometres.
3.
Convert feet to meters.
4.
Convert degrees Fahrenheit to degrees Celsius.
5.
Quit.
Enter an option (0 to see menu): 1
Enter miles: 10
This is equivalent to 16.1 kilometers
Enter an option (0 to see menu): 2
Enter number of feet: 1
This is equivalent to 0.31 meters
3
Enter an option (0 to see menu): 4
(12 marks)
3. (a)
A light beam of span 5m is simply supported at its endpoints. At the point x=3m there
is a load of 72kN. Express the Bending Moment M in terms of a step function. By
solving the differential equation
EI
d2y
=M
dx 2
y(0)=y(5)=0
find the deflection y at any point on the beam.
(b)
(9 marks)
A light beam of span 6m has both ends embedded in walls. Between the points x=2m
and x=6m there is a U.D.L. of 36kNm-1. Express the Bending Moment M in terms of
step function. Solve the differential equation
EI
d2y
=M
dx 2
to find the deflection y at any point on the beam.
(c)
(8 marks)
By using Euler’s Method or the Three Term Taylor Method with a step h=0.1 estimate
the value of y at x=1.1 where
dy
= 3y
dx
Note: y k+1 =y k +hy′k +
y(1)=2
h2
y′′k
2!
(4 marks)
Select any three of parts (a) to (d)
4.
(a)
Solve the differential equation
d2x
dx
+ 3 + 2x = 20
2
dt
dt
(b)
.
(6 marks)
Find the general solution of the differential equation
d 2 y dx
= 40cos4t
+2
dt 2
dt
(d)
(8 marks)
Find the general solution of the differential equation
d2x
dx
+ 2 + x = 4t
2
dt
dt
(c)
x(0)=x ′(0)=0 .
(7 marks)
Solve the differential equation
4
dx
− 2x = 4
dt
5. (a)
x(0)=0
(5 marks)
Find the first two non vanishing terms of a Maclaurin Series and the first three terms of a
Taylor Series expansion of f(x)=ln(secx) about the point x=
π
.
4
Note: 2cos2A=1+cos2A
1
cosA
sin2A=2sinAcosA
π
π
π
sin   = 1 tan   = 1 cos   = 0 cos 0 = 1
2
4
2
(b)
secA =
tan 0 = 0 .
(9 marks)
Find the Taylor series expansion of f(x,y)=exp(3x-y) about the values x=1,y=3.
Note: A Taylor Series expansion of f(x,y) about values x=a and y=b is given by
f(x, y)=f(a, b)+(x − a)f x +(y − b)f y +
(c)
(x − a) 2
(y − b) 2
f xx +(x − a)(y − b)f xy +
f yy + .... (6 marks)
2!
2!
In estimating a quantity V the formula
V=
2x
2x-y
is used. Find the partial derivatives of V with respect to x and to y. Hence estimate
the value of V where the values of x and y were measured as 2 and 3 with maximum
errors of 0.01 and 0.02, respectively.
6. (a)
Find the inverse of the matrix A below. If they exist find the matrices BC and CB.
1 1 3 


A=  1 2 1 
1 0 3 


(b)
(5 marks)
B= (1 2)
 1
C= 
 1
(9 marks)
Solve the set of equations below can be solved by using Gaussian Elimination with
partial
pivoting (correct to two places of decimal). Also solve this set of equations by using
Gaussian Elimination without partial pivoting or by using Cramers Rule.
5
 2 4 3  x   1 

   
 2 5 4  y  =  0 
 5 5 4  z   0 

   
7. (a)
(11 marks)
Variables R and T are related by a formula of the type R=a+bT. For the data below
by using the Least Squares Method find the best values of the constants a and b.
R
0.20
0.35
0.50
0.65
0.80
T
1
2
3
4
5
(7 marks
(b)
For the vectors a=2i+j+2k, b=i+2j+2k, c=i+j+k:
(i) Find the acute angle between the vectors a and b.
(ii) Verify that cx(axb)=(b.c)a-(a.c)b
(c)
(6 marks)
Two forces F1 and F2 act at the points A(1,0,-1). The force F1 is of magnitude
30kN and it acts in the direction of the vector AB where B is the points with
coordinates (3,-2,0).
The second force F2 is represented by the vector F2=10i+20j+30k.
(i) Find the vector F1.
(ii) Find the component of F2 in the direction of the vector AB.
(iii) Find the moments of both forces about the point B.
8. (a)
(7 marks)
(i) After production of items by a manufacturer it is found that 0.1% fail to satisfy certain
tolerances and these items are deemed to be defective.
By using the Binomial Distribution calculate the probability that a random sample
of 80 items contains two or more defective items.
(ii) In a computer laboratory on average each student sent three files to a printer during
any hour. By using the Poisson Distribution calculate the probability that a student
will send two or three files to the printer during any hour.
(b)
(7 marks)
The lengths of concrete blocks of a large batch are assumed to be Normally
distributed with a mean value of 450mm and with a standard deviation of 0.16mm.
What percentage of blocks have lengths between 450.3mm and 450.8mm?
If 99.8% of lengths are less than some critical value L find the value of L.
6
Calculate the probability that the mean length of a sample of 16 blocks will be greater
than 449.4mm.
(c)
(6 marks)
To monitor the breaking strength of precast concrete 6 samples of size 3were taken and
the measurements are recorded below .
Sample No
1
2
3
4
5
6
Mean
54.7
55.3
54.9
55.0
55.2
54.9
Range
0.5
0.4
0.5
0.4
0.4
0.5
Set up and plot a control chart for samples means. Include on the chart the mean
values of the samples above and comment on the control of the process.
.STANDARD DERIVATIVES & INTEGRALS
f ′(x)
f(x)
xn
sinx
nx n −1
cosx
cosx
-sinx
secx
secxtanx
tanx
sec2x
eax
aeax
dv
du
+v
u
dx
dx
du
dv
−u
v
dx
dx
2
v
uv
u
v
f(x)
∫ f(x)dx
1
x
f(x) = f(a) + (x − a)f ′(a) +
.BINOMIAL DISTRIBUTION
a=constant
x n +1
n +1
ln x
xn
TAYLOR SERIES
a=constant
(x − a) 2
f ′′(a) + ..
2!
 N
P(r)=   prqN-r
r 
7
where p+q=1
(7 marks)
POISSON DISTRIBUTION P(r)=
Normal Distribution
z=
x-µ
σ
λ r e −λ
where λ is the mean number of occurrences
r!
z=
x-µ
σ/ n
8